Universita degli Studi dellInsubria Quantum Monte Carlo Simulations
Universita’ degli Studi dell’Insubria Quantum Monte Carlo Simulations of Mixed 3 He/4 He Clusters Dario Bressanini dario. bressanini@uninsubria. it http: //www. unico. it/~dario
Overview n Introduction to quantum monte carlo methods ´ n Helium clusters simulations ´ n n VMC, QMC, advantages and drawbacks Problems, solutions Mixed 3 He/4 He clusters ´ Trimers ´ 3 He 4 He N Future directions © Dario Bressanini 2
Monte Carlo Methods n How to solve a deterministic problem using a Monte Carlo method? n Rephrase the problem using a probability distribution n “Measure” A by sampling the probability distribution © Dario Bressanini 3
Monte Carlo Methods n The points Ri are generated using random numbers This is why the methods are called Monte Carlo methods n Metropolis, Ulam, Fermi, Von Neumann (-1945) n We introduce noise into the problem!! ´ Our results have error bars. . . ´ . . . Nevertheless it might be a good way to proceed © Dario Bressanini 4
Quantum Mechanics n We wish to solve H Y = E Y to high accuracy ´ n The solution usually involves computing integrals in high dimensions: 3 -30000 The “classic” approach (from 1929): ´ Find approximate Y (. . . but good. . . ) ´ . . . whose integrals are analitically computable (gaussians) ´ Compute the approximate energy chemical accuracy ~ 0. 001 hartree ~ 0. 027 e. V © Dario Bressanini 6
VMC: Variational Monte Carlo n Start from the Variational Principle n Translate it into Monte Carlo language © Dario Bressanini 7
VMC: Variational Monte Carlo n E is a statistical average of the local energy EL over P(R) n Recipe: ´ take an appropriate trial wave function ´ distribute N points according to P(R) ´ compute the average of the local energy © Dario Bressanini 8
VMC: Variational Monte Carlo n There is no need to analytically compute integrals, so there is complete freedom in the choice of the trial wave function n Quantum chemistry uses a product of single particle functions. With VMC we can use any function: explicitly correlated wave functions can be used © Dario Bressanini r 12 r 2 He atom 9
The Metropolis Algorithm ? n How do we sample n Use the Metropolis algorithm (M(RT)2 1953). . . and a powerful computer Anyone who consider n The algorithm is a random arithmetical methods of producing random digits walk (markov chain) in is, of course, in a state of sin. configuration space John Von Neumann © Dario Bressanini 10
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The Metropolis Algorithm move Ri Call the Oracle Rtry reject accept Ri+1=Ri Ri+1=Rtry Compute averages © Dario Bressanini 12
The Metropolis Algorithm The Oracle if p 1 /* accept always */ accept move If 0 p 1 /* accept with probability p */ if p > rnd() accept move else reject move © Dario Bressanini 13
VMC advantages n No need to make the single-particle approximation n Can use for which no analytical integrals exist ´ Use explicitly correlated wave functions ´ Can satisfy the cusp conditions He atom ground state E 19 terms = -2. 9037245 a. u. Exact = -2. 90372437 a. u. © Dario Bressanini 14
VMC advantages n Can compute difficult quantities, e. g. n Can compute lower bounds © Dario Bressanini 15
VMC advantages n Can easily go beyond the Born-Oppenheimer approximation. H 2+ molecule ground state E 1 term = -0. 596235(9)a. u. E 10 terms = -0. 597136(3)a. u. Exact = -0. 597139 a. u. © Dario Bressanini 16
VMC advantages n Can work with ANY potential, in ANY number of dimensions. Ps 2 molecule (e+e+e-e-) in 2 D and 3 D n Optimization of nonlinear parameters ´ Numerically stable ´ Minimum known in advance (0) ´ Can be used for excited states with same symmetry too © Dario Bressanini 17
VMC drawbacks n Error bar goes down as N-1/2 n It is computationally demanding n The optimization of Y becomes difficult as the number of nonlinear parameters increases n It depends critically on our skill to invent a good Y n There exist exact, automatic ways to get better wave functions. Let the computer do the work. . . © Dario Bressanini 19
Diffusion Monte Carlo n VMC is a “classical” simulation method Nature is not classical, dammit, and if you want to make a simulation of nature, you'd better make it quantum mechanical, and by golly it's a wonderful problem, because it doesn't look so easy. Richard P. Feynman n Suggested by Fermi in 1945, but implemented only in the 70’s © Dario Bressanini 20
Diffusion equation analogy n The time dependent Schrödinger equation is similar to a diffusion equation n The diffusion equation can be “solved” by directly simulating the system Time evolution Diffusion Branch Can we do the same with the Schrödinger equation ? © Dario Bressanini 21
Imaginary Time Sch. Equation n The analogy is only formal ´ n Y is a complex quantity, while C is real and positive If we let the time t be imaginary, then Y can be real! Imaginary time Schrödinger equation © Dario Bressanini 22
Y as a concentration n Y is interpreted as a concentration of fictitious particles, called walkers n The schrödinger equation is simulated by a process of diffusion, growth and disappearance of walkers Ground State © Dario Bressanini 23
Diffusion Monte Carlo SIMULATION: discretize time • Diffusion process • Kinetic process (branching) © Dario Bressanini 24
The DMC algorithm © Dario Bressanini 25
The Fermion Problem n Wave functions for fermions have nodes. ´ Diffusion equation analogy is lost. Need to introduce positive and negative walkers. The (In)famous Sign Problem n Restrict random walk to a positive region bounded by nodes. Unfortunately, the exact nodes are unknown. n Use approximate nodes from a trial Y. Kill the walkers if they cross a node. © Dario Bressanini + - 26
Helium n A helium atom is an elementary particle. A weakly interacting hard sphere. n Interatomic potential is known more accurately than any other atom. ´Two isotopes: • 3 He (fermion: antisymmetric trial function, spin 1/2) • 4 He (boson: symmetric trial function, spin zero) • The interaction potential is the same © Dario Bressanini 27
Helium Clusters n They are not rigid, and show large-amplitude motion n Normal mode analysis useless n “Equilibrium structure” is ill-defined n Stochastic methods well suited to study helium clusters, both pure or with impurities ´ VMC, DMC, GFMC, PIMC etc. . . © Dario Bressanini 28
Helium Clusters 1. Small mass of helium atom 2. Very weak He-He interaction 0. 02 Kcal/mol 0. 9 * 10 -3 cm-1 0. 4 * 10 -8 hartree 10 -7 e. V Highly non-classical systems Superfluidity High resolution spectroscopy Low temperature chemistry © Dario Bressanini 30
The Simulations n Both VMC and DMC simulations n Potential = sum of two-body TTY pair-potential n Three-body terms not important for small clusters n Standard © Dario Bressanini 31
Pure 4 Hen Clusters The quality of the VMC simulations decreases as the cluster increases © Dario Bressanini 32
Y for 4 Hen Clusters n Wave function quality decreases as N increases ´ It was optimized to get minimum s(H), not minimum <H> ´ Are three- and many-body terms in Y important ? ´ Very difficult to optimize. Unstable process especially for the trimers. Can we improve Y ? © Dario Bressanini 33
Mixed 3 He/4 He Clusters (m, n) = 3 Hem 4 Hen Bressanini et. al. J. Chem. Phys. 112, 717 (2000) © Dario Bressanini 34
Helium Clusters: energy (cm-1) © Dario Bressanini 35
Helium Clusters: stability n 4 He is destabilized by substituting a 4 He with a 3 He N n The structure is only weakly perturbed. Dimers 4 He Bound Trimers 4 He 3 Bound Tetramers 4 He 4 Bound © Dario Bressanini 4 He 3 He 3 He Unbound 4 He 3 He 2 Bound Unbound 4 He 3 Bound 4 He 3 He 2 2 Bound 36
Trimers and Tetramers Stability 4 He -1 E = -0. 08784(7) cm 3 4 He 3 He E = -0. 00984(5) cm-1 2 Bonding interaction Non-bonding interaction 4 He -1 E = -0. 3 88 6 ( 1 ) cm 4 4 He 3 He E = -0. 2062(1) cm-1 3 4 He 3 He E = -0. 071(1) cm-1 2 2 Five out of six unbound pairs! © Dario Bressanini 37
3 He/4 He Distribution Functions Pair distribution functions 3 He(4 He) 5 © Dario Bressanini 38
3 He/4 He Distribution Functions Distributions with respect to the center of mass 3 He(4 He) 5 c. o. m © Dario Bressanini 39
Distribution Functions in 4 He. N 3 He r(4 He-4 He) © Dario Bressanini r(3 He-4 He) 40
Distribution Functions in 4 He. N 3 He c. o. m. = center of mass r(4 He-C. O. M. ) Similar to pure clusters © Dario Bressanini r(3 He-C. O. M. ) Fermion is pushed away 41
What is the shape of 4 He 3 ? n Some people say is an equilateral triangle. . . n . . . some say it is linear (almost). . . n . . . some say it is both. Pair distribution function © Dario Bressanini 42
What is the shape of 4 He 3 ? © Dario Bressanini 43
The Shape of the Trimers Ne trimer r(Ne-center of mass) He trimer r(4 He-center of mass) © Dario Bressanini 44
Ne 3 Angular Distributions a b Ne trimer b b a © Dario Bressanini a 45
4 He Angular Distributions 3 a b b b a © Dario Bressanini a 46
3 He 4 He Angular Distributions 2 a b b b a © Dario Bressanini a 47
Different wave function form Spline 0. 60 f(r) 0. 40 0. 20 0. 00 © Dario Bressanini 2. 00 4. 00 6. 00 r (a. u. ) 8. 00 10. 00 48
Y for 4 He 2 n Y literature (Rick & Doll) E = -0. 00046 cm-1 n Optimize s Unbound n Optimize E (numerically) E = -0. 00075 cm-1 n Y with Exp() E = -0. 00084 cm-1 n Y using splines E = -0. 00081 cm-1 n QMC E = -0. 00089(1) cm-1 n Numerical E = -0. 00091 cm-1 © Dario Bressanini 49
Y for 4 He 3 n Y literature (Rick & Doll) E = -0. 0798 cm-1 n Optimize Energy E = -0. 0829(4) cm-1 n Y with Exp() E = -0. 0851(2) cm-1 n Y using splines E = -0. 0868(2) cm-1 n QMC exact E = -0. 08784(7) cm-1 three-body terms are not important in Y for the trimer © Dario Bressanini 50
Work in Progress n Various impurities embedded in a Helium cluster n Y for bigger clusters using splines n Optimize the energy instead of the variance of the local energy. n Geometric structure of trimers and tetramers © Dario Bressanini 51
Conclusions n The substitution of a 4 He with a 3 He leads to an energetic destabilization. n 3 He weakly perturbes the 4 He atoms distribution. n 3 He moves on the surface of the cluster. n 4 He 3 He bound, 4 He 3 He unbound. 2 2 n 4 He 3 He and 4 He 3 He bound. 3 2 2 n QMC gives accurate energies and structural information © Dario Bressanini 52
Acknowledgments Gabriele Morosi Massimo Mella Mose’ Casalegno Giordano Fabbri Matteo Zavaglia © Dario Bressanini 53
A reflection. . . A new method for calculating properties in nuclei, atoms, molecules, or solids automatically provokes three sorts of negative reactions: Æ A new method is initially not as well formulated or understood as existing methods Æ It can seldom offer results of a comparable quality before a considerable amount of development has taken place Æ Only rarely do new methods differ in major ways from previous approaches Nonetheless, new methods need to be developed to handle problems that are vexing to or beyond the scope of the current approaches (Slightly modified from Steven R. White, John W. Wilkins and Kenneth G. Wilson) © Dario Bressanini 54
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